Isomonodromic quantization of dimensionally reduced gravity

Korotkin, Dmitry; Nicolai, Hermann

doi:10.1016/0550-3213(96)00320-3

Cited by 38 publications

(83 citation statements)

References 60 publications

(96 reference statements)

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“…& Remark 2.1. The relationship between the systems (2.13) and isomonodromic deformations is a generalization of the link between the Ernst equation and the Schlesinger system established by Korotkin & Nicolai (1996).…”

Section: (E ) Relationship To Isomonodromic Deformationsmentioning

confidence: 99%

“…For the isomonodromic sector, we relate the tau-function to the Jimbo-Miwa tau-function of the Schlesinger system. This link between two taufunctions generalizes the formula (found in Korotkin & Nicolai 1996) relating the so-called 'conformal factor' of the Ernst equation with the Jimbo-Miwa taufunction. We also show how to construct solutions of the systems (1.7) from solutions of matrix Riemann-Hilbert problems on CP 1 .…”

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confidence: 99%

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A new hierarchy of integrable systems associated to Hurwitz spaces

Kokotov

Korotkin

2007

Phil. Trans. R. Soc. A.

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In this paper, we introduce a new class of integrable systems, naturally associated to Hurwitz spaces (spaces of meromorphic functions over Riemann surfaces). The critical values of the meromorphic functions play the role of 'times'. Our systems give a natural generalization of the Ernst equation; in genus zero, they realize the scheme of deformation of integrable systems proposed by Burtsev, Mikhailov and Zakharov. We show that any solution of these systems in rank 1 defines a flat diagonal metric (Darboux-Egoroff metric) together with a class of corresponding systems of hydrodynamic type and their solutions.

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Section: (E ) Relationship To Isomonodromic Deformationsmentioning

confidence: 99%

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confidence: 99%

A new hierarchy of integrable systems associated to Hurwitz spaces

Kokotov

Korotkin

2007

Phil. Trans. R. Soc. A.

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“…Isomonodromic quantization of dimensionally reduced Gravity can be found in [57]. The relationship between W ∞ gravity (geometry) and the Fedosov deformation quantization of the 4D Self-Dual Gravity [38] associated with the complexified co-tangent space of a two-dim Riemann surface was studied by [40], [39].…”

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confidence: 99%

STRINGS AND MEMBRANES FROM EINSTEIN GRAVITY, MATRIX MODELS AND W_∞ GAUGE THEORIES AS PATHS TO QUANTUM GRAVITY

Castro

2008

Int. J. Mod. Phys. A

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It is shown how w∞, w1+∞ Gauge Field Theory actions in 2D emerge directly from 4D Gravity. Strings and Membranes actions in 2D and 3D originate as well from 4D Einstein Gravity after recurring to the nonlinear connection formalism of Lagrange-Finsler and Hamilton-Cartan spaces. Quantum Gravity in 3D can be described by a W∞ Matrix Model in D = 1 that can be solved exactly via the Collective Field Theory method. We describe why a quantization of 4D Gravity could be attained via a 2D Quantum W∞ gauge theory coupled to an infinite-component scalar-multiplet. A proof that non-critical W∞ (super) strings are devoid of BRST anomalies in dimensions D = 27 (D = 11), respectively, follows and which coincide with the critical (super) membrane dimensions D = 27 (D = 11). We establish the correspondence between the states associated with the quasi finite highest weights irreducible representations of W∞,W∞ algebras and the quantum states of the continuous Toda molecule. Schroedinger-like QM wave functional equations are derived and solutions are found in the zeroth order approximation. Since higher-conformal spin W∞ symmetries are very relevant in the study of 2D W∞ Gravity, the Quantum Hall effect, large N QCD, strings, membranes, ...... it is warranted to explore further the interplay among all these theories. In this introductory section we will review the work of [2] and afterwards we will discuss the recent work related the Hidden Symmetries of M theory. Keywords1.1 Gravity in D = m + n as an m-dim Gauge Theory of diffeomorphisms of an internal n-dim space and HolographySome time ago Park [1] showed that 4D Self Dual Gravity is equivalent to a WZNW model based on the group SU (∞). Namely, 4D Self Dual Gravity is the non-linear sigma model based in 2D whose target space is the "group manifold" of area-preserving diffs of another 2D-dim manifold. Roughly speaking, this means that the effective D = 4 manifold, where Self Dual Gravity is defined, is "spliced" into two 2D-submanifolds: one submanifold is the original 2D base manifold where the non-linear sigma model is defined. The other 2D submanifold is the target group manifold of area-preserving diffs of a two-dim sphere S 2 . The authors [2] went further and generalized this particular Self Dual Gravity case to the full fledged gravity in D = 2 + 2 = 4 dimensions, and in general, to any combinations of m + n-dimensions. Their main result is that m + n-dim Einstein gravity can be identified with an m-dimensional generally invariant gauge theory of Dif f s N , where N is an n-dim manifold. Locally the m + ndim space can be written as Σ = M × N and the metric G AB decomposes as: [62]. The decomposition (1.1) must not be confused with the Kaluza-Klein reduction where one imposes an isometry restriction on the γ AB that turns A a µ into a gauge connection associated with the gauge group G generated by isometry. Dropping the isometry restrictions allows all the fields to depend on all the coordinates x, y. Nevertheless A a µ (x, y) can still be identified as a conne...

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“…Although the complete related quantum model is still missing, Ref. [18] provides an exact quantization of the isomonodromic truncation of (2) in a covariant \two-time" framework. It would be highly interesting to recover traces of the mentioned…”

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confidence: 99%

“…We explicitly give the action of T (w) on the physical currents up to H-gauge transformations (3): (21) In particular, the set of g a 1 generates that half of the a ne algebra that acts nontrivially on the physical elds. Thus, having identi ed the action of g 1 among the transformations (18), the action of the entire nontrivial half ofmultiplet structures among the exact solutions of the Wheeler-DeWitt equation.…”

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confidence: 99%